On intermediate differentiability of Lipschitz functions on certain Banach spaces

Abstract

A real-valued function f defined on a Banach space X is said to be intermediately differentiable at x E X if there is 4 E X* such that for every h E X the value (4, h) lies between the upper and lower derivatives of f at x in the direction h . We show that if Y contains a dense continuous linear image of an Asplund space and X is a subspace of Y, then every locally Lipschitz function on X is generically intermediately differentiable. Let (X, 11 * 11) be a Banach space with dual X* and duality pairing between X* and X denoted by (., *). Recall that the upper and lower derivatives of a function f: X -R at x E X in a direction h E X are defined by D+f(x, h) = lim sup t[f(x + th) f(x)] tlO and D+f(x, h) = liminf l[f(x, th) f(x)] respectively. The function f is said to be intermediatly differentiable at x E X, with intermediate derivative 4 E X*, if D+f(x,h)>(4,h)>D+(x,h) forallheX. The aim of this note is to prove the following statement. Theorem. Suppose that a Banach space Y contains a dense continuous linear image of an Asplund space and that X is a subspace of Y. Then every locally Lipschitz function defined on an open subset Q of X is intermediately differentiable at every point of a residual subset of Q. The Banach spaces containing a dense continuous linear image of an Asplund space have been extensively studied by Ch. Stegall [11], who calls them GSG spaces. Among other things, he also proved [11, ?4, Remark 3] that the nonw.c.g. subspace of some LI (,u), ,u finite, constructed by H. Rosenthal [10] is Received by the editors April 23, 1990 and, in revised form, July 11, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G05; Secondary 46B22, 58C20.